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Wednesday, April 18, 2012

Communication is now the key to modelling distributed/multicore computations. Jim Demmel has been writing papers and giving talks on this theme for a while now, and as processors get faster, and the cloud becomes a standard computing platform, communication between nodes is turning out to be the major bottleneck.

So suppose you want to learn in this setting ? Suppose you have data sitting on different nodes (you have a data center, or a heterogeneous sensor network, and so on) and you'd like to learn something on the union of the data sets. You can't afford to ship everything to a single server for processing: the data might be too large to store, and the time to ship might be prohibitive.

We started out with the simplest of learning problems: classification. Supppose you have data sitting on two nodes (A and B), and you wish to learn a hypothesis over the union of A and B. What you'd like is a way for the nodes to communicate as little as possible with each other while still generating a hypothesis close to the optimal solution.

It's not hard to see that you could compute an $\epsilon$-sample on A, and ship it over to B. By the usual properties of an $\epsilon$-sample, you guarantee that any classifier on B's data combined with the sample will also classify A correctly to within some $\epsilon$-error. It's also not too hard to show a lower bound that matches this upper bound. The amount of communication is nearly linear in $1/\epsilon$.

But can you do better ? In fact yes, if you let the nodes talk to each other, rather than only allowing one-way communication. One way of gaining intuition for this is that $A$ can generate classifiers, and send them over to $B$, and $B$ can tell $A$ to turn the classifier left or right. Effectively, $B$ acts as an oracle for binary search. The hard part is showing that this is actually a decimation (in that a constant fraction of points are eliminated from consideration as support points in each step), and once we do that, we can show an exponential improvement over one-way communication. There's a trivial way to extend this to more than 2 players, with a $k^2$ blow up in communication for $k$ players.

This binary search intuition only works for points in 2D, because then the search space of classifiers is on the circle, which lends itself naturally to a binary search. In higher dimensions, we have to use what is essentially a generalization of binary search - the multiplicative weight update method. I'll have more to say about this in a later post, but you can think of the MWU as a "confused zombie" binary search, in that you only sort of know "which way to go" when doing the search, and even then points that you dismissed earlier might rise from the dead.

It takes a little more work to bring the overhead for k-players down to a factor k. This comes by selecting one node as a coordinator, and implementing one of the distributed continuous sampling techniques to pass data to the coordinator.

You can read the paper for more details on the method. One thing to note is that the MWU can be "imported" from other methods that use it, which means that we get distributed algorithms for many optimization problems for free. This is great because a number of ML problems essentially reduce to some kind of optimization.

A second design template is multipass streaming: it's fairly easy to see that any multipass sublinear streaming algorithm can be placed in the k-player distributed setting, and so if you want a distributed algorithm, design a multipass streaming algorithm first.

One weakness of our algorithms was that we didn't work in the "agnostic" case, where the optimal solution itself might not be a perfect classifier (or where the data isn't separable, to view it differently). This can be fixed: in an arxiv upload made simultaneously with ours, Blum, Balcan, Fine and Mansour solve this problem very neatly, in addition to proving a number of PAC-learning results in this model.

It's nice to see different groups exploring this view of distributed learning. It shows that the model itself has legs. There are a number of problems that remain to be explored, and I'm hoping we can crack some of them. In all of this, the key is to get from a 'near linear in error' bound to a 'logarithmic in error' bound via replacing sampling by active sampling (or binary search).

Monday, April 09, 2012

(tl;dr: Our upcoming paper in SoCG 2012 shows that with a nontrivial amount of work, you can do approximate Bregman near neighbor queries in low dimensional spaces in logarithmic query time)

One of the things that has haunted me over the years (yes, haunted!) is the geometry of Bregman divergences. A Bregman divergence is a very interesting creature. It's defined as the difference between a convex function and its linear approximation starting at another point:$$ d_\phi(p,q) = \phi(p) - \phi(q) - \langle \nabla \phi(q), p - q\rangle $$Because the Bregman divergences are parametrized by the function $\phi$, they yield a family of divergences. The most familiar one is $\ell_2^2$, but the most important one is the Kullback-Leibler divergence. There are a ton of reasons why one should study Bregman divergences, and in a shameless plug I'll refer to you my slides (one, two, three) from a geometry summer school last year. Suffice it to say that it's possible to unify a number of different algorithms in machine learning under a single framework by realizing that they're all particular instances of doing something with a Bregman divergence: two notable examples of this are Adaboost and information-theoretic clustering.

So what's the geometric perspective on Bregman divergences ? They generalize duality !

There's a standard way to think about traditional point-hyperplane duality via a lifting map: take a point, lift it to the paraboloid one dimension up, and find the tangent plane. But suppose we replace the paraboloid by a generic convex function ? what we get is a general convex duality (technically a Fenchel-Legendre duality) defined by$$\phi^*(u) = \sup_v \langle u,v \rangle - \phi(v)$$The reason this doesn't come up in Euclidean space is that the mapping is self-dual if we set $\phi(x) = (1/2) \|x\|^2$, which explains the symmetry in $\ell_2^2$.

It turns out that this observation is enough to import much of standard combinatorial geometry over to general Bregman spaces. We can compute convex hulls, Voronoi Diagrams, delaunay triangulations etc, as long as we're careful to keep primal and dual straight. for example, the locus of points equidistant from two points under a Bregman divergence is either a primal straight line or a dual straight line (depending on how you measure things), and so on. This was all elegantly worked out by Nielsen, Nock and Boissonnat a while ago.

Alas, this beautiful connection doesn't help us with approximation problems.

One of the most interesting questions regarding Bregman divergences is solving the near-neighbor problem. This is interesting because the Kullback-Leibler distance is often used (for example) to compare images, and so there's been a lot of empirical work on this problem.

But here's the problem. Let's consider the low dimensional ANN problem. In Euclidean space (or even in spaces of bounded doubling dimension), here's how you solve the problem. You build some kind of quad-tree-like data structure, using the triangle inequality to reason about which cells you need to explore, and using packing bounds to bound the number of cells explored. You also need a crude bound on the near neighbor to start with, and to do this, you use some variant of a ring-tree.

Bregman divergences don't even satisfy a directed triangle inequality in general. And to date, we didn't even know how to define packing bounds properly for these directed distances.

In an upcoming paper at SoCG 2012 with my students Amirali Abdullah and John Moeller, we finally figured out how to get some "approximation" of the ANN machinery to work with Bregman divergences, and get logarithmic query times with small space.

If I say so myself, there are some nice ideas in the paper. Firstly, in a somewhat surprising twist, a Bregman divergence satisfies a "reverse triangle inequality" on the line:

\[ d_\phi(a,b) + d_\phi(b,c) \le d_\phi(a, c), a, b, c, \in R\]

This is neat, because it gives packing bounds ! Intuitively, if the sum of lengths of subintervals along an interval is at most the length of the interval, then you can't have too many subintervals.

The next trick is an observation that the square root of a Bregman divergence satisfies a kind of relaxed triangle inequality that we call $\mu$-defectiveness:

\[ |d(x, y) - d(x,z)| \le \mu d(y, z)\]

This allows us to import some of the ring-tree machinery to get a coarse approximation.

And I should mention that the $\mu$ values involved here are quite small. If you're comparing distributions with the KL-divergence, then the value of $\mu$ is less than 2 even quite close to the boundary of the simplex.

Even with all of this in place, the quad-tree argument breaks. This is because of $\mu$-defectiveness: we can't assume that cell sizes are "reasonably large" at some number of levels below the root of the tree, and so our packing bounds look terrible. It takes a lot more work to fix this problem: essentially we exploit second-order structure of the Bregman divergences to bound the cell sizes and get the desired packing bound.

Afterthoughts:

Most of the complexity of the algorithm is in the analysis: the actual algorithm looks mostly like a Euclidean ANN procedure. While we haven't implemented it, I'm hopeful that when we do, we'll be able enjoy the empirical behavior of our Euclidean cousins.

We're looking at high dimensional Bregman near neighbors, as well as some other approximate Bregman geometry questions. While our low-D ANN result comes from the "I will break your limbs one by one till you submit" school of algorithms, the hope is that we can start to exploit more of the dual geometry as we learn more.

The goal of the project is to develop novel compressive sensing algorithms for
geoscience problems in the area of hydrocarbon field exploration and
surveillance. The appointment is initially for one-year, with the possibility
of renewal for up to 3 years. The appointment should start either during the
summer (the preferred option) or the fall of 2012.

Duties and Responsibilities:

Provide expertise on and contribute to the development of compressive
sensing and sparse recovery algorithms for geoscience applications

Help MIT faculty in coordination of research projects, including periodic
reporting and documentation as required by the program timeline